Abstract group $C_2^6.C_4$

• Label: 256.1535
• Order: \( 2^{8} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{16} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $3$

Group information

Description:	$C_2^6.C_4$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^9.C_2^3\wr C_2$, of order \(65536\)\(\medspace = 2^{16} \) (generators)
Outer automorphisms:	Group of order \(1024\)\(\medspace = 2^{10} \)
Composition factors:	$C_2$ x 8
Nilpotency class:	$3$
Derived length:	$2$

This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

Group statistics

Order	1	2	4	8
Elements	1	79	48	128	256
Conjugacy classes	1	23	8	8	40
Divisions	1	23	8	4	36
Autjugacy classes	1	7	2	1	11

Dimension	1	2	4
Irr. complex chars.	16	12	12	40
Irr. rational chars.	8	16	12	36

Minimal Presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$3$
Inequivalent generating triples:	$84$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{2}=f^{8}=[a,b]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,15,7,11,3,13,5,9)(2,16,8,12,4,14,6,10), (1,4)(2,3)(5,6)(7,8)(13,15)(14,16) \!\cdots\! \rangle$
Transitive group:	16T533	32T2644	32T2645	32T2646	all 8
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^5.C_4)$ $\,\rtimes\,$ $C_2$	$(C_2^3:\OD_{16})$ $\,\rtimes\,$ $C_2$	$C_2^3$ $\,\rtimes\,$ $(\OD_{16}:C_2)$	$(C_2^2:\OD_{16})$ $\,\rtimes\,$ $C_2^2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^6$ . $C_4$	$(C_2^2\times D_4)$ . $D_4$	$(C_2^4:D_4)$ . $C_2$	$(D_4\times C_2^3)$ . $C_4$	all 23

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 4247 subgroups in 1243 conjugacy classes, 87 normal (13 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^4:C_4$
Commutator:	$G' \simeq$ $C_2^4$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^3\times C_4$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6.C_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^6.C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^4:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 256: $C_2^6.C_4$

Order 128: $C_2^3:\OD_{16}$ x 4, $C_2^5.C_4$ x 2, $C_2^4:D_4$

Order 64: $C_2^3:D_4$ x 12, $C_2^3:C_8$ x 8, $\OD_{16}:C_2^2$ x 4, $C_2^2:\OD_{16}$ x 4, $D_4\times C_2^3$ x 3, $C_2^4:C_4$ x 3, $C_2^6$

Order 32: $C_2^2\times D_4$ x 32, $C_2^2\wr C_2$ x 32, $C_2^5$ x 22, $C_2^3:C_4$ x 22, $\OD_{16}:C_2$ x 8, $C_2^2:C_8$ x 8, $C_2\times \OD_{16}$ x 4, $C_2^3\times C_4$ x 3

Order 16: $C_2^4$ x 191, $C_2\times D_4$ x 100, $C_2^2:C_4$ x 24, $C_2^2\times C_4$ x 20, $\OD_{16}$ x 8, $C_2\times C_8$ x 4

Order 8: $C_2^3$ x 411, $D_4$ x 48, $C_2\times C_4$ x 28, $C_8$ x 4

Order 4: $C_2^2$ x 199, $C_4$ x 8

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $C_2^6.C_4$

Order 128: $C_2^5.C_4$, $C_2^4:D_4$, $C_2^3:\OD_{16}$

Order 64: $D_4\times C_2^3$ x 2, $C_2^3:D_4$ x 2, $C_2^4:C_4$ x 2, $\OD_{16}:C_2^2$, $C_2^2:\OD_{16}$, $C_2^3:C_8$, $C_2^6$

Order 32: $C_2^5$ x 7, $C_2^2\times D_4$ x 7, $C_2^3:C_4$ x 5, $C_2^3\times C_4$ x 2, $\OD_{16}:C_2$, $C_2^2:C_8$, $C_2\times \OD_{16}$, $C_2^2\wr C_2$

Order 16: $C_2^4$ x 24, $C_2\times D_4$ x 9, $C_2^2\times C_4$ x 7, $C_2^2:C_4$ x 2, $\OD_{16}$, $C_2\times C_8$

Order 8: $C_2^3$ x 37, $C_2\times C_4$ x 7, $D_4$ x 2, $C_8$

Order 4: $C_2^2$ x 24, $C_4$ x 2

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $C_2^6.C_4$ ($C_1$)

Order 128: $C_2^3:\OD_{16}$ ($C_2$) x 4, $C_2^5.C_4$ ($C_2$) x 2, $C_2^4:D_4$ ($C_2$)

Order 64: $C_2^2:\OD_{16}$ ($C_2^2$) x 4, $D_4\times C_2^3$ ($C_4$) x 2, $C_2^4:C_4$ ($C_2^2$) x 2, $C_2^6$ ($C_4$), $D_4\times C_2^3$ ($C_2^2$), $C_2^4:C_4$ ($C_4$)

Order 32: $C_2^3:C_4$ ($D_4$) x 8, $C_2^5$ ($C_2\times C_4$) x 4, $C_2^2\times D_4$ ($D_4$) x 4, $C_2^3\times C_4$ ($C_2\times C_4$) x 2, $C_2^3\times C_4$ ($C_2^3$)

Order 16: $C_2^4$ ($C_2^2:C_4$) x 8, $C_2^2\times C_4$ ($C_2\times D_4$) x 6, $C_2^2\times C_4$ ($C_2^2:C_4$) x 4, $C_2^4$ ($C_2^2\times C_4$)

Order 8: $C_2^3$ ($\OD_{16}:C_2$) x 8, $C_2^3$ ($C_2^3:C_4$) x 4, $C_2\times C_4$ ($C_2^2\wr C_2$) x 4, $C_2^3$ ($C_2^3:C_4$) x 3

Order 4: $C_2^2$ ($\OD_{16}:C_2^2$) x 4, $C_2^2$ ($C_2^4:C_4$) x 2, $C_2^2$ ($C_2^4:C_4$)

Order 2: $C_2$ ($C_2^5.C_4$) x 2, $C_2$ ($C_2^5:C_4$)

Order 1: $C_1$ ($C_2^6.C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $C_2^6.C_4$ ($C_1$)

Order 128: $C_2^5.C_4$ ($C_2$), $C_2^4:D_4$ ($C_2$), $C_2^3:\OD_{16}$ ($C_2$)

Order 64: $C_2^2:\OD_{16}$ ($C_2^2$), $C_2^6$ ($C_4$), $D_4\times C_2^3$ ($C_2^2$), $D_4\times C_2^3$ ($C_4$), $C_2^4:C_4$ ($C_2^2$), $C_2^4:C_4$ ($C_4$)

Order 32: $C_2^5$ ($C_2\times C_4$) x 3, $C_2^2\times D_4$ ($D_4$), $C_2^3\times C_4$ ($C_2^3$), $C_2^3\times C_4$ ($C_2\times C_4$), $C_2^3:C_4$ ($D_4$)

Order 16: $C_2^4$ ($C_2^2:C_4$) x 3, $C_2^2\times C_4$ ($C_2\times D_4$) x 2, $C_2^4$ ($C_2^2\times C_4$), $C_2^2\times C_4$ ($C_2^2:C_4$)

Order 8: $C_2^3$ ($C_2^3:C_4$) x 2, $C_2^3$ ($\OD_{16}:C_2$), $C_2^3$ ($C_2^3:C_4$), $C_2\times C_4$ ($C_2^2\wr C_2$)

Order 4: $C_2^2$ ($\OD_{16}:C_2^2$), $C_2^2$ ($C_2^4:C_4$), $C_2^2$ ($C_2^4:C_4$)

Order 2: $C_2$ ($C_2^5.C_4$), $C_2$ ($C_2^5:C_4$)

Order 1: $C_1$ ($C_2^6.C_4$)

Series

Derived series

$C_2^6.C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_2^6.C_4$

$\rhd$

$C_2^4:D_4$

$\rhd$

$C_2^4:C_4$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^6.C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3\times C_4$

$\lhd$

$C_2^6.C_4$

Supergroups

This group is a maximal subgroup of 16 larger groups in the database.

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

See the $40 \times 40$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $36 \times 36$ rational character table.